The House of Representatives has impeached Donald Trump for inciting insurrection at the Capitol when Congress was counting electoral votes. The Senate may convict him of that charge. If it does, it will decide whether to disqualify him from “any office of honor, trust, or profit under the United States,” which likely (I think) includes the office of the president.

There are important reasons for the Senate to consider convicting and disqualifying Mr. Trump, and the stigma of disqualification might prevent a future candidacy. But this penalty isn’t a magic trick that would make Mr. Trump disappear from the political stage. It wouldn’t bar him from fundraising for a future presidential candidacy or from appearing on the ballot in a future election. And the question may ultimately return to Congress one day when it counts electoral votes.

The Constitution requires that the president must be at least 35 years old, a natural born citizen, and a resident of the United States for 14 years. There are other tacit requirements, too—for example, one must be alive instead of dead, and a person instead of, say, a dog or a cat. Disqualification from office after impeachment would be another condition.

These eligibility requirements are usually self-policing. Underage or non-citizen candidates rarely attempt to run for president, and major political parties winnow out ineligible candidates. Other disputes over eligibility, like whether Canadian-born Senator Ted Cruz was a “natural born citizen,” never became serious problems because candidates lost elections.

A candidate who runs for president must file with the Federal Election Commission to disclose campaign contribution and expenditure data. But the FEC doesn’t have power to determine whether candidates are eligible.

In 2011, a naturalized citizen, Abdul Hassan, sought to run for president. He admitted he wasn’t a natural born citizen but asked the FEC if he could still run. An advisory legal opinion from the FEC concluded that campaign finance law allowed him to solicit funds for his campaign, and that he wouldn’t be engaging in fraudulently misrepresentation if he did so.

Any candidate, eligible or ineligible, can run a presidential campaign in the United States. Ineligible candidates would not violate any campaign finance laws by soliciting financial contributions or running for president. Congress would need to amend campaign finance laws to bar ineligible candidates from fundraising for office.

Candidates running for office must also file paperwork in states to assure their names appear on the primary and general election ballots. Many states do not investigate the qualifications of candidates seeking elected office. They trust candidates, voters, and political parties to make those judgments and act appropriately.

It’s up to states to decide whether to enact laws to exclude disqualified candidates. But ineligible presidential candidates do sometimes appear on the ballot. Róger Calero, for instance, is a Nicaraguan who resides in the United States and was the Socialist Workers Party candidate for president in 2004 and 2008. He’s not a citizen, much less a natural born citizen, but he appeared on the ballot in New Jersey, New York, and other states in two elections. Or consider Peta Lindsay, who was just 28 years of age when she ran as the Socialism and Liberation Party nominee in 2012 and earned 7791 votes across nine states.

Some states do exclude unqualified presidential candidates. For instance, Colorado requires candidates to affirm under oath that they are qualified. In 2012, Mr. Hassan couldn’t affirm that, and he sued. Then-Judge Neil Gorsuch wrote a judicial opinion affirming the state’s right to exclude Mr. Hassan from appearing on the ballot.

But others have tended toward mischief. After false rumors swirled in 2008 that Barack Obama was not born in the United States, some state legislatures introduced legislation that would require candidates to show their birth certificate as a condition of appearing on the ballot. None became law.

In the event a disqualified candidate appeared on the ballot, he might receive electoral votes from a state, and those electoral votes would be sent to Congress to count. In 1873, Congress refused to count three electoral votes from Georgia cast for Horace Greeley, a candidate who died after Election Day but before the electors met. Congress rejected votes cast for an ineligible candidate.

Under the Electoral Count Act, a member of the House and a member of the Senate may object to counting votes that were not “regularly given.” Both houses of Congress would then need to agree to reject the votes cast for an ineligible candidate.

That means Congress might ultimately be forced to evaluate Mr. Trump’s eligibility when it counted electoral votes.

Yes, during that meeting.